Speaker | Transcript |
| Our talk today shows how to |
| use JMP to do Bayesian |
| estimation. Here's an overview |
| of my talk. I'm going to start |
| with a brief introduction to |
| Bayesian statistical methods. |
| Then I'm going to go through |
| four different examples that |
| happen to come from reliability, |
| but the methods we're presenting |
| are really much more general and |
| can be applied in other areas of |
| application. Then I'm going to |
| turn it over to Peng and he's |
| going to show you how easy it is |
| to actually do these things in |
| JMP. Technically, reliability |
| is a probability. The |
| probability of a system, |
| vehicle, machine or whatever it |
| is that is of interest, will |
| perform its intended function |
| under encountered operating |
| conditions for a specified period |
| of time. I highlight encountered |
| here to emphasize that |
| reliability depends importantly |
| on the environment in which a |
| product is being used. Condra |
| defined reliability as quality |
| over time. And many engineers |
| think of reliability is being |
| failure avoidance, that is, to |
| design and manufacture a product |
| that will not fail. Reliability |
| is a highly quantitative |
| engineering discipline, but |
| often requires sophisticated |
| statistical and probabilistic |
| ideas. Over the past 30 years, |
| there's been a virtual |
| revolution where Bayes methods |
| are now commonly used and in |
| many different areas of |
| application. This revolution |
| started by the rediscovery of |
| Markov chain Monte Carlo methods |
| and was accelerated by |
| spectacular improvements in |
| computing power that we have |
| today, as well as the |
| development of relatively easy |
| to use software to implement |
| Bayes methods. In the 1990s we |
| had BUGS. Today Stan and other |
| similar packages have largely |
| replaced BUGS, but the other |
| thing that's happening is we're |
| beginning to see more Bayesian |
| methods implemented in |
| commercial software. So for |
| example, SAS has PROC MCMC. |
| And now JMP has some very |
| powerful tools that were |
| developed for reliability, but |
| as I said, they can be applied |
| in other areas as well, and |
| there's strong motivation for |
| the use of Bayesian methods. |
| For one thing, it provides a |
| means for combining prior |
| information with limited data to |
| be able to make useful |
| inferences. Also, there are many |
| situations, particularly with |
| random effects complications |
| like censor data, where maximum |
| likelihood is difficult to |
| implement, but where Bayes |
| methods are relatively easy to |
| implement. There's one little |
| downside in the use of Bayes |
| methods. You have to think a bit |
| harder about certain things, |
| particularly about |
| parameterization and how to |
| specify the prior distributions. |
| My first example is about an |
| aircraft engine bearing cage. |
| These are field failure data |
| where there was 1,703 aircraft |
| engines that contained this |
| bearing cage. The oldest ones |
| had 2,220 hours of operation. The |
| design life specification for |
| this bearing cage was that no |
| more than 10% of the units would |
| fail by 8,000 hours of operation. |
| However, 6 units had failed and |
this raised the question | do we |
| have a serious problem here? Do |
| we need to redesign this bearing |
| cage to meet that reliability |
| condition? This is an event plot |
| of the data. The event plot |
| illustrates the structure |
| of the data, and in particular, |
| we can see the six failures |
| here. In addition to that, we |
| have right censored |
| observations, indicated here by |
| the arrows. So these are units |
| that are still in service and |
| they have not failed yet, and |
| the right arrow indicates that |
| all we know is if we wait long |
| enough out to the right, the units |
| will eventually fail. Here's a |
| maximum likelihood analysis of |
| those data, so the probability |
| plot here suggests that the |
| Weibull distribution provides a |
| good description of these data. |
| However, when we use the |
| distribution profiler to |
| estimate fraction failing at |
| 8,000 hours, we can see that the |
| confidence interval is enormous, |
| ranging between about 3% all the |
| way up to 100%. That's not very |
| useful. So likelihood methods |
| work like this. We specify the |
| model and the data. |
| That defines the likelihood and |
| then we use the likelihood to |
| make inferences. Bayesian |
| methods are similar, except we |
| also have prior information |
| specified. Bayes theorem |
| combines the likelihood with the |
| prior information, providing a |
| posterior distribution, and then |
| we use the posterior |
| distribution to make inferences. |
| Here's the Bayes analysis of the |
| bearing cage. The priors are |
| specified here for the B10 |
| or time at which 10% would fail. |
| We have a very wide interval |
| here. The range, effectively 1,000 |
| hours up to 50,000 hours. |
| Everybody would agree that B10 |
| is somewhere in that range. For |
| the Weibull shape parameter, |
| however, we're going to use an |
| informative prior distribution |
| based upon the engineers' |
| knowledge of the failure |
| mechanism and their vast |
| previous experience with that |
| mechanism. They can say with |
| little doubt that the Weibull |
| shape parameter should be |
| between 1.5 and 3, and |
| here's where we specify that |
| information. So instead of |
| specifying information for the |
| traditional Weibull parameters, |
| we've reparameterized, where now |
| the B10 is one of the |
| parameters, and here's the |
| specified range. And then we |
| have the informative prior for |
| the Weibull shape parameter |
| specified here. And then JMP |
| will generate samples from the |
| joint posterior, leading to |
| these parameter estimates and |
| confidence intervals shown here. |
| Here's a graphical depiction of |
| the Bayes analysis. The black |
| points here are a sample from |
| the prior distribution, so again |
| very wide for the .1 quantile |
| and somewhat constrained for the |
| Weibull shape parameter beta. On |
| the right here, we have the |
| joint posterior, which in effect |
| is where the likelihood contours |
| and the prior sample |
| overlap. And then those draws |
| from the joint posterior are |
| used to compute estimates and |
| Bayesian credible intervals. So |
| here's the same profiler that we |
| saw previously where the |
| confidence interval was not |
| useful. After bringing in the |
| information about the Weibull |
| shape parameter, now we can see |
| that the confidence interval |
| ranges between 12% and 83%, |
| clearly illustrating that we |
| have missed the target of 10%. |
| So what have we learned here? |
| With a small number of failures, |
| there's not much information |
| about reliability. But engineers |
| often have information that can |
| be used, and by using that prior |
| information, we can get improved |
| precision and more useful |
| inferences. And Bayesian methods |
| provide a formal method for |
| combining that prior information |
| with our limited data. Here's |
| another example. Rocket motor is |
| one of five critical components |
| in a missile. In this particular |
| application, there were |
| approximately 20,000 missiles |
| in the inventory. Over time, 1,940 |
| of these missiles had been fired |
| and they all worked, except in |
| three cases, where there was |
| catastrophic failure. And these |
| were older missiles, and so |
| there was some concern that |
| there might be a wearout failure |
| mechanism that would put into |
| jeopardy the roughly 20,000 |
| missiles remaining in inventory. |
| The failures were thought to be |
| due to thermal cycling, but |
| there was no information about |
| the number of thermal cycles. We |
| only have the age of the |
| missile when it was fired. |
| That's a useful surrogate, but |
| the effect of using a surrogate |
| like that is you have more |
| variability in your data. Now |
| in this case, there were no |
| directly observed failure times. |
| When a rocket is called upon to |
| operate and it operates |
| successfully, all we know is |
| that they had not failed at the |
| age of those units when they |
| were asked to operate. And for |
| the units that failed |
| catastrophically, again, we |
| don't know the time that those |
| units failed. At some point |
| before they were called upon to |
| operate. They had failed, so all |
| we know is that the failure was |
| before the age at which it was |
| fired. So as I said, there was |
| concern that there is a wear out |
| failure mechanism kicking in |
| here that would put into |
| jeopardy the amount of remaining |
| life for the units in the |
| stockpile. So here's the table |
| of the data. Here we have the |
| units that operated |
| successfully, and so these are |
| right censored observations, but |
| these observations here |
| are the ones that failed and as |
| I said, at relatively higher |
| ages. This is the event plot of |
| the data and again, we can see |
| the right censored observations |
| here with the arrow pointing to |
| the right, and we can see the |
| left censored observations with |
| the arrow pointing to the left |
| indicating the region of |
| uncertainty. But even with those |
| data we can still fit a Weibull |
| distribution. And here's the |
| probability plot showing the |
| maximum likelihood estimate and |
| confidence bands. Here's more |
| information from the maximum |
| likelihood analysis. And here we |
| have the estimate of fraction |
| failing at 20 years, which was |
| the design life of these rocket |
| motors, and again the interval |
| is huge, ranging between 3% and |
| 100%. Again, not very useful. |
| But the engineers, |
| knowing what the failure |
| mechanism was, again had |
| information about the Weibull |
| shape parameter. The maximum |
| likelihood estimate was |
| extremely large and the |
| engineers were pretty sure that |
| that was wrong, especially with |
| the extra variability in the |
| data that would tend to drive |
| the Weibull shape parameter to a |
| lower value. As I showed you on |
| the previous slide, confidence |
| interval for fraction failing at |
| 20 years was huge. So once again, |
| we're going to specify a prior |
| distribution and then use that |
| in a Bayes analysis. Again, the |
| prior for B10, the time at which |
| 10% will fail, is chosen to be |
| extremely wide. We don't really |
| want to assume anything there, |
| and everybody would agree that |
| that quantity is somewhere |
| between five years and 400 |
| years. But for the Weibull shape |
| parameter, we're going to assume |
| that it's between one and five. |
| Again, we know it's greater than |
| one because it's a wear out |
| failure mechanism, and the |
| engineers were sure that it |
| wasn't anything like the number |
| 8 that we had seen in the |
| maximum likely estimate. |
| And indeed, five is also a very |
| large Weibull shape parameter. |
| Once again, JMP is called upon |
| to generate draws from the |
| posterior distribution. And here are |
| plot similar to the ones that we |
| saw in the bearing cage example. |
| The black points here, again, are |
| a sample from the prior |
| distribution. Again very wide in |
| terms of the B10, but somewhat |
| constrained for the beta, so the |
| beta is an informative prior |
| distribution. And again the |
| contour plots represent the |
| information in the limited data. |
| In our posterior, once again, is |
| where we get overlap between the |
| prior and the likelihood and we |
| can see it here. So once again |
| we have a comparison between the |
| maximum likelihood interval, |
| which is extremely wide, and the |
| interval that we get for the |
| same quantity using the Bayes |
| inference, which incorporated the |
| prior information on the Weibull |
| shape parameter. And now the |
| interval ranges between .002 and |
| .98 or about .1, about 10% |
| failing, so that might be |
| acceptable. Some of the things |
| that we learned here, even |
| though there were no actual |
| failure times, we can still get |
| reliability information from the |
| data, but with very few failures |
| there isn't much information |
| there. But we can use the |
| engineer's knowledge about the |
| Weibull shape parameter to |
| supplement the data to get |
| useful inferences and JMP makes |
| this really easy to do. My last |
| two examples are about |
| accelerated testing. Accelerated |
| testing is a widely used |
| technique to get information |
| about reliability of components |
| quickly when designing a |
| product. The basic idea is to |
| test units at high levels of |
| variables like temperature or |
| voltage to make things fail |
| quickly and then to use a model |
| to extrapolate back down to the |
| use conditions. Extrapolation is |
| always dangerous and we have to |
| keep that in mind. That's the |
| reason we would like to have our |
| model be physically motivated. |
| So here's an example of an |
| accelerated life test |
| on a laser. Units were tested at |
| 40, 60 and 80 degrees C, but the |
| use condition was 10 degrees C. |
| That's the nominal temperature |
| at the bottom of the Atlantic |
| Ocean, where these lasers were |
| going to be used in a new |
| telecommunications system. The |
| test lasted 5,000 hours, a little |
| bit more than six months. The |
| engineers wanted to estimate |
| fraction failing at about 30,000 |
| hours. That's about 3.5 years |
| and again, at 10 degrees C. |
| Here's the results of the |
| analysi. In order to |
| appropriately test and build the |
| model, JMP uses these three |
| different analyses. The first |
| one fits separate distributions |
| to each level of temperature. |
| The next model does the same |
| thing, except that it constrains |
| the shape parameter Sigma to be |
| the same at every level of |
| temperature. This is analogous |
| to the constant Sigma assumption |
| that we typically make in |
| regression analysis. And then |
| finally, we fit the regression |
| model, which in effect, is a |
| simple linear regression |
| connecting lifetime to |
| temperature. And to supplement |
| this visualization of these |
| three models, JMP does |
| likelihood ratio tests to test |
| whether there's evidence that |
| the Sigma might depend on |
| temperature and then to test |
| whether there's evidence of lack |
| of fit in the regression model. |
| And from the large P values |
| here, we can see that there's no |
| evidence against this model. |
| Another way to plot the results |
| of fitting this model |
| is to plot lifetime versus |
| temperature on special scales. A |
| log rhythmic scale for hours of |
| operation in what's known as an |
| Arrhenius scale for temperature. |
| Corresponding to the Arrhenius |
| model, which describes how |
| temperature affects reaction |
| rates, and thereby lifetime. And |
| this is the results of the |
| maximum likelihood estimation |
| for our model. The JMP |
| distribution profiler gives us |
| an estimate of the fraction |
| failing at 30,000 hours. |
| And we can see it ranges between |
| .002 and about .12, or 12% |
| failing. The engineers in |
| applications like this, however, |
| often have information about |
| what's known as the effective |
| activation energy of the failure |
| mechanism, and that corresponds |
| to the slope of the regression |
| line in the Arrhenius model. So |
| we did a Bayes analysis and in |
| that analysis, we made an |
| assumption about the effective |
| activation energy. And that's |
| going to provide more precision |
| for us. So what we have here is |
| a matrix scatterplot of the |
| joint posterior distribution |
| after having specified prior |
| distributions for the |
| parameters, weakly informative |
| for the .1 quantile at 40 |
| degrees C. Again, everybody |
| would agree that that number is |
| somewhere between 100 and |
| 32,000. Also weakly informative |
| for the lognormal shape |
| parameter. Again, everybody |
| would agree that that number is |
| somewhere between .05 |
| and 20. But for the slope of the |
| regression line, we have an |
| informative prior ranging |
| between .6 and .8, based upon |
| previous experience with the |
| failure mechanism. And that leads |
| to this comparison, where now |
| on the right-hand side here, the |
| interval for fraction failing at |
| 30,000 hours is much narrower |
| than it was with the maximum |
| likelihood estimate. In |
| particular, the upper bound now |
| is only about 4% compared with |
| 12% for the maximum likelihood |
| estimates. So lessons learned. |
| Accelerated tests provide |
| reliability information quickly, |
| and engineers often have |
| information about the effect of |
| activation energy. And that can |
| be used to either improve |
| precision or to reduce cost by |
| not needing to test so many |
| units. And once again, Bayesian |
| methods provide an appropriate |
| method to combine the engineers' |
| knowledge with the limited data. |
| My final example concerns an |
| accelerated life test of |
| an integrated circuit device. Units |
| were tested at high temperature |
| and the resulting data were |
| interval censored. That's |
| because failures were discovered |
| only during inspections that |
| were conducted periodically. In |
| this test, however, there were |
| only failures at the two high |
| levels of temperature. The goal |
| of the test was to estimate the |
| .01 quantile at 100 degrees C. |
| This is a table of the data |
| where we can see the failures at |
| 250 and 300 degrees C. |
| And no failures all right |
| censored at the three lower |
| levels of temperature. Now when |
| we did the maximum likelihood |
| estimation, in this case, we saw |
| strong evidence that the Weibull |
| shape parameter depended on |
| temperature. So the P value is |
| about .03. That turns out |
| to be evidence against the |
| Arrhenius model, and that's |
| because the Arrhenius model should |
| only scale time. But if you |
| change the shape parameter by |
| increasing temperature, you're |
| doing more than scaling time. |
| And so that's a problem, and it |
| suggested that at 300 degrees C, |
| there was a different failure |
| mechanism. And indeed, when the |
| engineers followed up and |
| determined the cause of failure |
| of the units at 250 and 300, |
| they saw that there was a |
| different mechanism at 300. What |
| that meant is that we had to |
| throw those data away. So what |
| do we do then? Now we've only |
| got failures at 250 degrees C |
| and JMP doesn't do very well |
| with that. It's surprisingly, |
| actually runs and gives |
| answers, but the confidence |
| intervals are enormously wide |
| here, as one would expect. But |
| the engineers knew what the |
| failure mechanism was and they |
| had had previous experience and |
| so they can bring that |
| information about the slope into |
| the analysis using Bayes |
| methods. So again, here's the |
| joint posterior and the width of |
| the distribution in the |
| posterior for beta 1 is |
| effectively what we assumed |
| when we put in a prior |
| distribution for that parameter. |
| So again, here's the |
| specification of the prior |
| distributions, where we used |
| weakly informative for the |
| quantile and for Sigma, but |
| informative prior distribution |
| for the slope beta 1. And I can |
| get an estimate of the time at |
| which 1% fail. So the lower end |
| point of the confidence interval |
| for the time at which 1% will fail |
| is more than 140,000 hours. |
| So that's about 20 years, much |
| longer than the technological |
| life of these products in which |
| this integrated circuit will be |
| used. So what did we learn here? |
| Well, in some applications we |
| have interval censoring because |
| failures are discovered only |
| when there's an inspection. We |
| need appropriate statistical |
| methods for handling such data, |
| and JMP has those methods. If |
| you use excessive levels of an |
| accelerating variable like |
| temperature, you can generate |
| new failure modes that make the |
| information misleading. So we |
| had to throw those units away. |
| But even with failures at only |
| one level of temperature, if we |
| have prior information |
| about the effective activation |
| energy, we can combine that |
| information with the limited |
| data to make useful inferences. |
| Finally, some concluding |
| remarks, improvements in |
| computing hardware and software |
| have greatly advanced our ability |
| to analyze reliability and other |
| data. Now we can also use Bayes |
| methods, providing another set of |
| tools for combining information |
| with limited data and JMP has |
| powerful tools for doing this. |
| So, although these Bayesian |
| capabilities were developed for |
| the reliability part of JMP, |
| they can certainly be used in |
| other areas of application. And |
| here are some references, |
| including the 2nd edition of |
| Statistical Methods Reliability, |
| which should be out probably in |
| June of 2021. OK, so now I'm |
| going to turn it over to Peng |
| and he's going to show you how |
| easy it is to do these analyses. |
| Thank you, professor. |
| Before I start my |
| demonstration, I would like |
| to show this slide about |
| Bayesian analysis workflow |
| in life distribution and |
| Fit Life by X. |
| First, you need to fit a |
| parametric model using maximum |
| likelihood. I assume you already |
| know how to do this in these two |
| platforms. Then you need to |
| review or find model |
| specification graphical user |
| interface for Bayesian |
| estimation within the report |
| from the previous step. For |
| example, this is screenshot of |
| Weibull model in Life |
| Distribution. You need to go to the |
| red triangle menu |
| and choose Bayesian estimates |
| to reveal the graphical |
| user interface for the |
| Bayesian analysis. |
| In Fit Life by X, please see |
| the screenshot of a lognormal |
| result. And the graphical user |
| interface for the Bayesian |
| inference is on the last step. |
| After finding the graphical user |
| interface for Bayesian analysis, |
| you will need to supply the |
| information about the priors. |
| You need to decide the priors |
| dispersion for individual |
| parameters. You need to supply |
| the information for the |
| hyperparameters and additional |
| information such as the |
| probability for the quantile. |
| In addition to that, we |
| need to provide the number |
| of posterior samples. |
| Also need to provide a random |
| seed in case you want to |
| replicate your result in the |
| future. Then you can click Fit |
| Model. This will generate a |
| report of the model. |
| You can fit multiple models |
| in case you want to study the |
| sensitivity of different |
| Bayesian models given |
| different prior distribution. |
| The result of a Bayesian model, |
| including the following things, |
| first is a method of sampling. |
| And then is a copy of the priors. |
| Then had a posterior estimates |
| of the parameters. |
| And then there are some scatterplots, |
| either for the prior or |
| the posteriors. |
| In the end, we have two |
| profilers. One for distribution |
| and the one for the quantile. |
| Using these results, you |
| can make further inferences |
| such as failure prediction. |
| Now look at the demonstration. |
| We will demonstrate with the |
| last example that the professor |
| mentioned that ??? presentation. |
| It's the IC device there. |
| We have two columns for the time |
to event | HoursL and HoursU to |
| represent the censoring situation. |
| We have a count |
| for individual observation and |
| temperature for individual |
| observation. We exclude the last |
| four observations because they |
| are associated with a different |
| failure mode. We want to exclude |
| these observations from the |
| analysis. Now start to specify. |
| ???. |
| We put hours into Y |
| We put Count into frequency. |
| We put Degrees C into X. |
| We use the Arrhenius Celsius |
| for our relationship. |
| We use lognormal for our |
| distribution. |
| Then click OK. |
| The result is the maximum |
| likelihood influence |
| for the lognormal. We go to the |
| Bayesian estimates, and |
| start to specify our priors like |
| Professor did in his |
| presentation. We choose a |
| lognormal for the quantile. |
| It's 250 degrees C. |
| And its B10 life. So |
| probability is 0.1. |
| The two ends of the |
| lognormal distribution is 100 |
| and 10,000. |
| Now specify the slope. |
| Distribution is lognormal. |
| And two ends of the |
| distribution is .65 |
| and .85 because it's informative |
| to require the range is narrow. |
| Now we specify the prior |
| distribution for Sigma, |
| which is a lognormal |
| and it had a wide range; |
| it's .05 and 5. |
| We decided to draw 5,000 |
| posterior samples. And assign an |
| arbitrary random seed. And then |
| we click...click on Fit Model. |
| And what the report generates for this |
| specification. The method is |
| simple rejection. And here's a |
| copy of our proir specification. |
| The posterior estimates |
| summarize our posterior samples. |
| You can export the posterior samples |
| by either clicking this Export |
| Monte Carlo samples |
| or choose it from the menu, |
which is here | Export Monte |
| Carlo Samples. |
| Posterior samples are illustrated |
| in these scatterplots. |
| We have two scatterplots here. |
| The first to use |
| the same paramaterization as the prior |
| specification, which is...which |
| use quantile, slope and signal. |
| The other scatter plot....the |
| second scatter plot use a |
| traditional parameterization, |
| which includes the intercept of |
| regression, a slope of the |
| regression and Sigma. |
| In the end to make ???, we |
| can you we can look at the |
| profiler. Here let's look |
| at the second profiler, |
| quantile profile, so we can |
| find the same result |
| as what Professor had shown in |
| one of the previous slides. |
| Enter 0.1... |
| 0.01 for probability. So this is |
| 1%. We enter 100 degree C |
| for DegreesC. |
| And we adjust |
| the axes. |
| So now we see a similar |
| profiler. |
| It has that was already in |
| the previous slide. |
| And we can read off the Y axis |
| to get the result we want, which |
| is the time that 1% of the |
| device will fail at 100 degrees |
| C. So this concludes my |
| demonstration. And let |
| me move on. |
| This slide explains |
| about...explains JMP implementation |
| of sampling algorithm. |
| We have seen that the simple |
| rejection has shown up in the |
| previous example and this is the |
| first stage of our |
| implementation. The simple |
| rejection algorithm is tried and |
| true method to your samples, |
| but it can be impractical |
| if rejection rate is high. |
| So if the rejection |
| rate is high, we...our |
| implementations switch to the |
| second stage, which is a Random |
| Walk Metropolis-Hastings |
| algorithm. The second algorithm |
| is efficient, but in case...in |
| situations it can fail |
| undetectably if the likelihood |
| is irregular. For example, the |
| likelihood is rather flat. We |
| designed this implementation |
| because we have a situation, |
| there are very few failures or |
| even no failures. In that |
| situation the likelihood is |
| relatively high, but ??? |
| situation, we use simple |
| rejection algorithm and the |
| rejection rate is not that bad |
| and this method will suffice. |
| When we have more and more |
| failures, the likelihood it |
| becomes more regular. So it has |
| a peak in the middle. In that |
| situation, the simple |
| rejecction rate...the simple |
| rejection method becomes |
| impractical because of the high |
| rejection rate. But the Random |
| Walk algorithm becomes more and |
| more promising to succeed |
| without failure. So this is our |
| implementation and explanation of |
| why we do that. |
| This slide explains how do we |
| specify truncated normal prior |
| in these two platforms. Because |
| truncated normal is not a |
| building prior distribution in |
| these two platforms. |
| First, look at what is |
| truncated normal. Here we |
| give example of truncated normal |
| with two ends at 5 and 400. The |
| two ends are illustrated by |
| this L and this R. |
| But truncated normal is nothing |
| but a normal by discarding |
| all the values that are |
| negative, which is represented |
| by this through curve as the |
| equivalent normal distribution |
| for this particular truncated |
| normal distribution. |
| If we want to specify this |
| truncated normal or we need to |
| define is equivalent...equivalent |
| normal distribution with two ends |
| that had that had the same new |
| and Sigma parameters as those of |
| these truncated normal |
| distributions. So we provide a |
| script to do this. |
| In this script, the |
| calculation is the |
| following. We find the new |
| and Sigma from the truncated |
| normal. |
| So we get can get the |
| equivalent normal |
| distribution. |
| And we that ??? Sigma of this |
| normal distribution we can find |
| out the two ends of this normal |
| distribution. |
| So to specify the truncated |
| normal with this two end |
| value to specify equivalent |
| normal distribution with two |
| end...these two end values. |
| This is how we specify |
| truncated normal in these |
| two platforms using |
| equivalent normal |
| distribution. And here's the |
| content of the script. |
| All you need to do is by calling |
| a function, which here is the |
| reverse parameter tnorm to |
| normal value. What do you need |
| to provide are the two ends of |
| the truncated normal |
| distribution and it will give |
| the two ends of the equivalent |
| normal distribution and you can |
| use those two numbers to specify |
| the prior distribution. |
| So this concludes my |
| demonstration. In my |
| demonstration I showed how to |
| start Bayesian analysis in Life |
| distribution and Fix Life by |
| X, how to enter prior |
| information, |
| and what's the content of |
| Bayesian result. Also I explained |
| what our implementation of the |
| sampling and why do you do |
| that. And in the end I explain |
| how do we specify a truncated |
| normal prior using an |
| equivalent normal prior in |
| these two...in these two |
| platforms. Thank you. |